Often you need a notation for a finite sequence with one element is removed; i.e. $$(x_1,\dots,x_{i-1},x_{i+1}\dots, x_n).$$ I know one notation $$(x_1,\dots,\hat x_i,\dots, x_n)$$ and I hate it. It is too long and it has no sense; i.e., unless you know the meaning you will never guess what is it.
Question: Did you see any other?
In game theory, such sequences are needed all the time, and the notation $x_{-i}$ has become so common that it is often not even defined in papers.
The reason is that much of game theory is concerned with situations where each player $j$ has a presupposed strategy $x_j$ and we think of one player $i$ deviating from his given strategy to some other strategy $y_i$, while the other players do not deviate. This new outcome is often denoted by $(y_i,x_{-i})$ or $(y_i; x_{-i})$ or some such abuse of notation, instead of the cumbersome $(x_1,\ldots, x_{i-1},y_i,x_{i+1},\ldots,x_n)$. Despite the fact that "the indices are out of order," it is very convenient notation for game theory once you get used to it.
In particular it allows one to write conditions like $u_i(x)\geq u_i(y_i,x_{-i})$ for all players $i$ and all $y_i$ to define what it means for $x$ to be a Nash equilibrium. Other solution concepts can also be defined compactly with this notation.
How about $x|_{[n]\setminus\{i\}}$ ?
Borrowing from simplicial sets/complexes, $d_i: \mathbf{n-1} \to \mathbf{n}$ is the map that skips $i$, so your sequences would be $x\circ d_i$.
For any $n\in\mathbb{N}$ and for any $i\in[n]:=\{1,\dots,n\}$, you may consider the maps $\tau_i:[n-1] \to [n]$ defined in Iverson notation by $$\tau_i(x):=x+[x\ge i]\, .$$ That is, $\tau_i(x)=x$ unless $x\ge i$, in which case it is $x+1$). It induces by composition the map $\tau_i^*:\mathbb{N}^n\to\mathbb{N}^{n-1}$ that takes the element $x\in \mathbb{N}^n$ to $x\circ\tau_i$, which is what you want. In case of need, to recall the domain we may write $\tau_{i,n}$ instead of $\tau_i$; also, to simplify the notation, $\tau_i\cdot x$ instead of $\tau_i^*(x)$.
This and similar notations are somehow useful e.g. in treating technicalities with the constructions in singular homology. You may write down a list of simple identities relating e.g. compositions of these simple maps, and the analogous identities obtained by counter-functoriality on the $^*$-maps.
